Must the prayers performed while wearing socks that were washed in an automatic washing machine with other clothes be repeated, after having worn them in the airplane lavatory and considering them impure, even though the website states the necessity of washing impure clothes before putting them in the washing machine?

The default state of the bathroom floor and anything that touches it, such as socks and slippers, is purity, unless one is certain that it has been contaminated with impurity. Therefore, the socks with which one entered the airplane bathroom remain pure and are not considered impure merely due to doubt. The prayer performed with them is valid and does not need to be repeated. There is no harm in washing socks with clothes in the same washing machine as long as their purity is affirmed.